Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group

Abstract

We consider the asymptotic behavior as n→∞ of the spectra of random matrices of the form $$\frac{1}{\sqrt{n-1}}\sum_{k=1}^{n-1}Z_{nk}\rho_{n}\bigl ((k,k+1)\bigr),$$ where for each n the random variables Z_nk are i.i.d. standard Gaussian and the matrices ρ_n((k, k+1)) are obtained by applying an irreducible unitary representation ρ_n of the symmetric group on {1, 2, …, n} to the transposition (k, k+1) that interchanges k and k+1 [thus, ρ_n((k, k+1)) is both unitary and self-adjoint, with all eigenvalues either +1 or −1]. Irreducible representations of the symmetric group on {1, 2, …, n} are indexed by partitions λ_n of n. A consequence of the results we establish is that if λ_n,1≥λ_n,2≥⋯≥0 is the partition of n corresponding to ρ_n, μ_n,1≥μ_n,2≥⋯≥0 is the corresponding conjugate partition of n (i.e., the Young diagram of μ_n is the transpose of the Young diagram of λ_n), lim_n→∞λ_n,i/n=p_i for each i≥1, and lim_n→∞μ_n,j/n=q_j for each j≥1, then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean θZ and variance 1−θ², where θ is the constant ∑_ip_i²−∑_jq_j² and Z is a standard Gaussian random variable.